Some variants of the elimination method

Having stipulated condition (63), the computational formulae of the right elimination with regard to problem (61)-(62) can be written as 

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Substituting yi = yi i + Wj+i/aj+i from (65) into the first formula (64) yields 

To make our exposition more transparent, we introduce a = Oj (1 — dj) and ji = a-i (3i. With these, equation (8) admits the simplified form 

By comparing the first boundary condition (67) with (69) for i = 1 we find that 

A new unknown difference function is called a flow and is defined by 

We note in passing that formulae (71)-(72) just established are very flexible and more convenient for computational procedures in the case a 1. 

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As mentioned in the Introduction, one of our purposes is reduction of the non-Hermitian character of the ordinary EOMCC formalism (or, in some variants of the new theory, its complete elimination). Unlike the original Hamiltonian Hk, the transformed Hamiltonian H used in the EOMCC method is no longer Hermitian and the non-Hermiticity of H measured by a difference H — (H) shows up already in the first-order of MBPT, i.e. 

When feature selection is used to simplify, because of the large number of variables, methods must be simple. The univariate criterion of interclass variance/intraclass variance ratio (in the different variants called Fisher weights variance weights or Coomans weights is simple, but can lead to the elimination of variables with some discriminant power, either separately or, more important, in connection with other variables (Fig. 36). 

The reaction of CINO with F has been used to determine F atom concentrations by mass spectrometric measurement of the CIF produced after the addition of excess CINO. The ClF ion peak was calibrated by the addition of measured quantities of ClF from a cylinder (14). To eliminate the need to work with the reactive CIF gas, a variant of this method was developed in which the F atoms were titrated with ClNO to an endpoint identified as the amount of added CINO beyond which the CIF peak no longer increased ( ). The CINO methods are less well established them some other, inasmuch as the rate constant of reaction (1) has not been determined. Also, at high concentrations, the reaction NO + F2 -> FNO + F, which has a rate constant of 8 x 10 cm molecule" s at 300 K, may occur to an apprecicible extent, leading to an erroneous value for fJ. 

The first syntheses of dendralenes by C2-C3 bond formation (Scheme 1.25) were reported by Tsuge and coworkers in 1985 and 1986, and proceed via substitution at either a bromide 160 or an epoxide 163, followed by elimination (Scheme 1.26) [116, 117]. Similar addition/elimination sequences to carbonyl groups or epoxides [120], and substitution reactions [121], followed. Such methods have been superseded by cross-coupling techniques that take place between a 2-functionalized 1,3-butadiene and an alkene (each can be either electrophilic or nucleophilic) or a 4-functionalized 1,2-butadiene and alkene, and occur with allylic transposition (Scheme 1.25). No doubt due to the ready availability of alkenyl halides and allenes, and the variety of increasingly mild and selective reaction variants, cross-coupling has provided access to a large number of diversely substituted dendralenes over the past 20 years, some of which have even been part of natural product syntheses [14,122,123]. 

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